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21 pages, 3512 KiB

Open AccessEditor’s ChoiceArticle

Explicit Stable Finite Difference Methods for Diffusion-Reaction Type Equations

by Humam Kareem Jalghaf, Endre Kovács, János Majár, Ádám Nagy and Ali Habeeb Askar

Mathematics 2021, 9(24), 3308; https://doi.org/10.3390/math9243308 - 19 Dec 2021

Cited by 14 | Viewed by 2856

By the iteration of the theta-formula and treating the neighbors explicitly such as the unconditionally positive finite difference (UPFD) methods, we construct a new 2-stage explicit algorithm to solve partial differential equations containing a diffusion term and two reaction terms. One of the [...] Read more.

By the iteration of the theta-formula and treating the neighbors explicitly such as the unconditionally positive finite difference (UPFD) methods, we construct a new 2-stage explicit algorithm to solve partial differential equations containing a diffusion term and two reaction terms. One of the reaction terms is linear, which may describe heat convection, the other one is proportional to the fourth power of the variable, which can represent radiation. We analytically prove, for the linear case, that the order of accuracy of the method is two, and that it is unconditionally stable. We verify the method by reproducing an analytical solution with high accuracy. Then large systems with random parameters and discontinuous initial conditions are used to demonstrate that the new method is competitive against several other solvers, even if the nonlinear term is extremely large. Finally, we show that the new method can be adapted to the advection–diffusion-reaction term as well. Full article

(This article belongs to the Special Issue Application of Iterative Methods for Solving Nonlinear Equations)

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15 pages, 283 KiB

Open AccessArticle

Convergence Criteria of Three Step Schemes for Solving Equations

by Samundra Regmi, Christopher I. Argyros, Ioannis K. Argyros and Santhosh George

Mathematics 2021, 9(23), 3106; https://doi.org/10.3390/math9233106 - 2 Dec 2021

Cited by 1 | Viewed by 1180

Abstract

We develop a unified convergence analysis of three-step iterative schemes for solving nonlinear Banach space valued equations. The local convergence order has been shown before to be five on the finite dimensional Euclidean space assuming Taylor expansions and the existence of the sixth [...] Read more.

We develop a unified convergence analysis of three-step iterative schemes for solving nonlinear Banach space valued equations. The local convergence order has been shown before to be five on the finite dimensional Euclidean space assuming Taylor expansions and the existence of the sixth derivative not on these schemes. So, the usage of them is restricted six or higher differentiable mappings. But in our paper only the first Frèchet derivative is utilized to show convergence. Consequently, the scheme is expanded. Numerical applications are also given to test convergence. Full article

(This article belongs to the Special Issue Application of Iterative Methods for Solving Nonlinear Equations)

8 pages, 234 KiB

Open AccessArticle

A Note on Traub’s Method for Systems of Nonlinear Equations

by Beny Neta

Mathematics 2021, 9(23), 3073; https://doi.org/10.3390/math9233073 - 29 Nov 2021

Cited by 2 | Viewed by 983

Abstract

Traub’s method was extended here to systems of nonlinear equations and compared to Steffensen’s method. Even though Traub’s method is only of order 1.839 and not quadratic, it performed better in the 10 examples. Full article

(This article belongs to the Special Issue Application of Iterative Methods for Solving Nonlinear Equations)

15 pages, 284 KiB

Open AccessArticle

Extended Kung–Traub Methods for Solving Equations with Applications

by Samundra Regmi, Ioannis K. Argyros, Santhosh George, Ángel Alberto Magreñán and Michael I. Argyros

Mathematics 2021, 9(20), 2635; https://doi.org/10.3390/math9202635 - 19 Oct 2021

Viewed by 1349

Abstract

Kung and Traub (1974) proposed an iterative method for solving equations defined on the real line. The convergence order four was shown using Taylor expansions, requiring the existence of the fifth derivative not in this method. However, these hypotheses limit the utilization of [...] Read more.

Kung and Traub (1974) proposed an iterative method for solving equations defined on the real line. The convergence order four was shown using Taylor expansions, requiring the existence of the fifth derivative not in this method. However, these hypotheses limit the utilization of it to functions that are at least five times differentiable, although the methods may converge. As far as we know, no semi-local convergence has been given in this setting. Our goal is to extend the applicability of this method in both the local and semi-local convergence case and in the more general setting of Banach space valued operators. Moreover, we use our idea of recurrent functions and conditions only on the first derivative and divided difference, which appear in the method. This idea can be used to extend other high convergence multipoint and multistep methods. Numerical experiments testing the convergence criteria complement this study. Full article

(This article belongs to the Special Issue Application of Iterative Methods for Solving Nonlinear Equations)

16 pages, 5739 KiB

Open AccessArticle

A New High-Order Jacobian-Free Iterative Method with Memory for Solving Nonlinear Systems

by Ramandeep Behl, Alicia Cordero, Juan R. Torregrosa and Sonia Bhalla

Mathematics 2021, 9(17), 2122; https://doi.org/10.3390/math9172122 - 1 Sep 2021

Viewed by 1481

Abstract

We used a Kurchatov-type accelerator to construct an iterative method with memory for solving nonlinear systems, with sixth-order convergence. It was developed from an initial scheme without memory, with order of convergence four. There exist few multidimensional schemes using more than one previous [...] Read more.

We used a Kurchatov-type accelerator to construct an iterative method with memory for solving nonlinear systems, with sixth-order convergence. It was developed from an initial scheme without memory, with order of convergence four. There exist few multidimensional schemes using more than one previous iterate in the very recent literature, mostly with low orders of convergence. The proposed scheme showed its efficiency and robustness in several numerical tests, where it was also compared with the existing procedures with high orders of convergence. These numerical tests included large nonlinear systems. In addition, we show that the proposed scheme has very stable qualitative behavior, by means of the analysis of an associated multidimensional, real rational function and also by means of a comparison of its basin of attraction with those of comparison methods. Full article

(This article belongs to the Special Issue Application of Iterative Methods for Solving Nonlinear Equations)

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19 pages, 1358 KiB

Open AccessArticle

A New Nonlinear Ninth-Order Root-Finding Method with Error Analysis and Basins of Attraction

by Sania Qureshi, Higinio Ramos and Abdul Karim Soomro

Mathematics 2021, 9(16), 1996; https://doi.org/10.3390/math9161996 - 20 Aug 2021

Cited by 10 | Viewed by 1895

Abstract

Nonlinear phenomena occur in various fields of science, business, and engineering. Research in the area of computational science is constantly growing, with the development of new numerical schemes or with the modification of existing ones. However, such numerical schemes, objectively need to be [...] Read more.

Nonlinear phenomena occur in various fields of science, business, and engineering. Research in the area of computational science is constantly growing, with the development of new numerical schemes or with the modification of existing ones. However, such numerical schemes, objectively need to be computationally inexpensive with a higher order of convergence. Taking into account these demanding features, this article attempted to develop a new three-step numerical scheme to solve nonlinear scalar and vector equations. The scheme was shown to have ninth order convergence and requires six function evaluations per iteration. The efficiency index is approximately 1.4422, which is higher than the Newton’s scheme and several other known optimal schemes. Its dependence on the initial estimates was studied by using real multidimensional dynamical schemes, showing its stable behavior when tested upon some nonlinear models. Based on absolute errors, the number of iterations, the number of function evaluations, preassigned tolerance, convergence speed, and CPU time (sec), comparisons with well-known optimal schemes available in the literature showed a better performance of the proposed scheme. Practical models under consideration include open-channel flow in civil engineering, Planck’s radiation law in physics, the van der Waals equation in chemistry, and the steady-state of the Lorenz system in meteorology. Full article

(This article belongs to the Special Issue Application of Iterative Methods for Solving Nonlinear Equations)

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15 pages, 308 KiB

Open AccessArticle

Unified Convergence Criteria for Iterative Banach Space Valued Methods with Applications

by Ioannis K. Argyros

Mathematics 2021, 9(16), 1942; https://doi.org/10.3390/math9161942 - 14 Aug 2021

Cited by 42 | Viewed by 2034

Abstract

A plethora of sufficient convergence criteria has been provided for single-step iterative methods to solve Banach space valued operator equations. However, an interesting question remains unanswered: is it possible to provide unified convergence criteria for single-step iterative methods, which are weaker than earlier [...] Read more.

A plethora of sufficient convergence criteria has been provided for single-step iterative methods to solve Banach space valued operator equations. However, an interesting question remains unanswered: is it possible to provide unified convergence criteria for single-step iterative methods, which are weaker than earlier ones without additional hypotheses? The answer is yes. In particular, we provide only one sufficient convergence criterion suitable for single-step methods. Moreover, we also give a finer convergence analysis. Numerical experiments involving boundary value problems and Hammerstein-like integral equations complete this paper. Full article

(This article belongs to the Special Issue Application of Iterative Methods for Solving Nonlinear Equations)

13 pages, 295 KiB

Open AccessArticle

Some High-Order Convergent Iterative Procedures for Nonlinear Systems with Local Convergence

by Ramandeep Behl, Ioannis K. Argyros and Fouad Othman Mallawi

Mathematics 2021, 9(12), 1375; https://doi.org/10.3390/math9121375 - 14 Jun 2021

Cited by 2 | Viewed by 1614

Abstract

In this study, we suggested the local convergence of three iterative schemes that works for systems of nonlinear equations. In earlier results, such as from Amiri et al. (see also the works by Behl et al., Argryos et al., Chicharro et al., Cordero [...] Read more.

In this study, we suggested the local convergence of three iterative schemes that works for systems of nonlinear equations. In earlier results, such as from Amiri et al. (see also the works by Behl et al., Argryos et al., Chicharro et al., Cordero et al., Geum et al., Guitiérrez, Sharma, Weerakoon and Fernando, Awadeh), authors have used hypotheses on high order derivatives not appearing on these iterative procedures. Therefore, these methods have a restricted area of applicability. The main difference of our study to earlier studies is that we adopt only the first order derivative in the convergence order (which only appears on the proposed iterative procedure). No work has been proposed on computable error distances and uniqueness in the aforementioned studies given on

R^{k}

. We also address these problems too. Moreover, by using Banach space, the applicability of iterative procedures is extended even further. We have examined the convergence criteria on several real life problems along with a counter problem that completes this study. Full article

(This article belongs to the Special Issue Application of Iterative Methods for Solving Nonlinear Equations)

8 pages, 331 KiB

Open AccessArticle

On High-Order Iterative Schemes for the Matrix pth Root Avoiding the Use of Inverses

by Sergio Amat, Sonia Busquier, Miguel Ángel Hernández-Verón and Ángel Alberto Magreñán

Mathematics 2021, 9(2), 144; https://doi.org/10.3390/math9020144 - 11 Jan 2021

Cited by 2 | Viewed by 1471

Abstract

This paper is devoted to the approximation of matrix pth roots. We present and analyze a family of algorithms free of inverses. The method is a combination of two families of iterative methods. The first one gives an approximation of the matrix [...] Read more.

This paper is devoted to the approximation of matrix pth roots. We present and analyze a family of algorithms free of inverses. The method is a combination of two families of iterative methods. The first one gives an approximation of the matrix inverse. The second family computes, using the first method, an approximation of the matrix pth root. We analyze the computational cost and the convergence of this family of methods. Finally, we introduce several numerical examples in order to check the performance of this combination of schemes. We conclude that the method without inverse emerges as a good alternative since a similar numerical behavior with smaller computational cost is obtained. Full article

(This article belongs to the Special Issue Application of Iterative Methods for Solving Nonlinear Equations)

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15 pages, 415 KiB

Open AccessArticle

A Picard-Type Iterative Scheme for Fredholm Integral Equations of the Second Kind

by José M. Gutiérrez and Miguel Á. Hernández-Verón

Mathematics 2021, 9(1), 83; https://doi.org/10.3390/math9010083 - 1 Jan 2021

Cited by 3 | Viewed by 2061

Abstract

In this work, we present an application of Newton’s method for solving nonlinear equations in Banach spaces to a particular problem: the approximation of the inverse operators that appear in the solution of Fredholm integral equations. Therefore, we construct an iterative method with [...] Read more.

In this work, we present an application of Newton’s method for solving nonlinear equations in Banach spaces to a particular problem: the approximation of the inverse operators that appear in the solution of Fredholm integral equations. Therefore, we construct an iterative method with quadratic convergence that does not use either derivatives or inverse operators. Consequently, this new procedure is especially useful for solving non-homogeneous Fredholm integral equations of the first kind. We combine this method with a technique to find the solution of Fredholm integral equations with separable kernels to obtain a procedure that allows us to approach the solution when the kernel is non-separable. Full article

(This article belongs to the Special Issue Application of Iterative Methods for Solving Nonlinear Equations)

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